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A functional law of the iterated logarithm for multi-class queues with batch arrivals

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Abstract

A functional law of the iterated logarithm (LIL) and its corresponding LIL are established for a multiclass single-server queue with first come first served (FCFS) service discipline. The functional LIL and its LIL quantify the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. The functional LIL and LIL are established in three cases: underloaded, critically loaded and overloaded, for performance measures including the total workload, idle time, queue length, workload, busy time, departure and sojourn time processes. The proofs of the functional LIL and LIL are based on a strong approximation approach, which approximates discrete performance processes with reflected Brownian motions. Numerical examples are considered to provide insights on these limit results.

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Acknowledgements

The authors thank the editor and the anonymous reviewer for their guidance and constructive comments. The first author is supported by the National Nature Science Foundation of China under Grant Nos. 11871116 and 11971074 and the Fundamental Research Funds for the Central Universities of China under Grant No. 2019XD-A11.

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Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People’s Republic of China

    Yongjiang Guo

  2. School of Management, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Xiyang Hou

  3. Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, USA

    Yunan Liu

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  1. Yongjiang Guo
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  2. Xiyang Hou
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  3. Yunan Liu
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Correspondence to Yunan Liu.

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Guo, Y., Hou, X. & Liu, Y. A functional law of the iterated logarithm for multi-class queues with batch arrivals. Ann Oper Res 300, 51–77 (2021). https://doi.org/10.1007/s10479-020-03864-6

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